Lesson 1.1

1.1 Proof by mathematical induction Quiz: Pearson Edexcel Further Maths, Unit 1

20 questions

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Lesson 1.1, Proof by mathematical induction: 20 multiple choice questions for the Pearson Edexcel Further Maths (9FM0), Unit 1: Proof, written with Revision Ninja.

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The 20 questions

  1. What is the main purpose of the base case in mathematical induction?

    • Starting the chain
    • Assuming truth
    • Finding the limit
    • Proving the step

  2. What is the name given to the assumption that the statement is true for n = k in an induction proof?

    • The contrapositive
    • The converse statement
    • The inductive hypothesis
    • The base case

  3. Which statement is the correct pattern for the sum of the first n odd numbers?

    • 1 + 3 + 5 + ... + (2n - 1) = n(n + 1)
    • 1 + 3 + 5 + ... + (2n - 1) = n^2
    • 1 + 3 + 5 + ... + (2n - 1) = n^3
    • 1 + 3 + 5 + ... + (2n - 1) = 2n^2

  4. A statement is to be proved by induction for all integers n >= 2. At which value must the base case be checked?

    • n = 3
    • n = 1
    • n = 0
    • n = 2

  5. What is the value of the sum of r^2 from r = 1 to 1?

    • 2
    • 0
    • 1
    • 6

  6. Why does testing finite values of n fail as a full mathematical proof?

    • Calculators make errors
    • Negative numbers exist
    • Invalid algebra
    • Infinitely many cases

  7. What is the value of sum of r(r + 1) from r = 1 to 4?

    • 36
    • 30
    • 40
    • 50

  8. For sum of r(r + 1) from r = 1 to n equal to n(n + 1)(n + 2)/3, what must be added to the sum when moving from n = k to n = k + 1?

    • (k + 1)
    • (k + 1)(k + 2)
    • k(k + 1)
    • (k + 2)(k + 3)

  9. For M = [[3, -4], [1, -1]], what is the (1,2) entry of M squared?

    • 8
    • -3
    • -8
    • -4

  10. Which expression is divisible by 6 for every positive integer n?

    • 7^n + 1
    • 7^n - 3
    • 7^n - 1
    • 7^n - 2

  11. In proving 6 divides 7^n - 1 by induction, which expression for 7^(k+1) - 1 shows divisibility by 6 once 6 divides 7^k - 1 is assumed?

    • 7(7^k + 1) - 6
    • 6 x 7^k + 1
    • 7(7^k - 1) - 6
    • 7(7^k - 1) + 6

  12. What is the value of both sides of 2^n >= n + 1 when n = 0?

    • 1
    • 0.5
    • 0
    • 2

  13. Multiplying both sides of 2^k >= k + 1 by 2 gives which expression?

    • 2^(k+1) >= 2k + 2
    • 2^k + 1 >= 2k + 2
    • 2^(k+1) >= k + 2
    • 2^(k+1) >= 2k + 1

  14. What is the sum of the cubes from r = 1 to 3?

    • 45
    • 36
    • 24
    • 27

  15. If the sum of cubes from r = 1 to n equals n^2(n + 1)^2/4, what is the sum when n = 4?

    • 100
    • 64
    • 120
    • 90

  16. For the statement that k^3 + 2k is divisible by 3, which expansion correctly shows (k+1)^3 + 2(k+1) = (k^3 + 2k) + 3(k^2 + k + 1)?

    • (k^3 + 2k) + 3(k^2 + 2k + 1)
    • (k^3 + 2k) + (k^2 + k + 1)
    • (k^3 + 2k) + 3(k^2 + k + 1)
    • (k^3 + 2k) + 3(k^2 + k)

  17. In the matrix induction for M^n = [[2n+1, -4n], [n, 1-2n]], which expression gives the (2,2) entry of M^(k+1) = M x M^k?

    • -(4k) - 2k, which simplifies to -6k
    • -4k + (1 - 2k), which simplifies to 1 - 6k
    • 1 - 2k - 1, which simplifies to -2k
    • -4k - (1 - 2k), which simplifies to -2k - 1 = 1 - 2(k+1)

  18. What key component is missing if a proof only tests the base case?

    • Inductive step
    • Counterexample
    • Base case
    • Expanded form

  19. If P(1) is true and P(k) implies P(k + 2), which numbers are proven?

    • Odd positive integers
    • All prime numbers
    • Even positive integers
    • All positive integers

  20. What does strong induction assume true for the step at k?

    • Only base case
    • Only case k
    • All previous cases
    • All positive integers

All Pearson Edexcel Further Maths quizzes